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Convergence and summable almost \(T\)-stability of the random Picard-Mann hybrid iterative process. (English) Zbl 1351.47051
Let \((\Omega ,\xi ,\mu )\) be a probability space, \(E\) a Banach space, \(T:\Omega \times E\to E\) a random operator, and \(x^*\) the random fixed point of \(T\). The authors study parametrized versions of the Mann and the Picard-Mann hybrid iterative processes for the approximation of \(x^*\). They establish convergence and summable almost stability of these iteration schemes for the operator \(T\) satisfying the following contraction-type condition: \[ \| x^*(\omega )-T(\omega ,y)\| \leq \theta (\omega )\| x^*(\omega )-y\|-\varphi (\| x^*(\omega )-y\| , \] where \(\omega \in\Omega \), \(y\in E\), \(0\leq \theta (\omega )<1\), \(\varphi \) is nondecreasing and \(\varphi (t)>0\) for \(t>0\).
Reviewer’s remark: It is not difficult to see that a simpler version of the above condition, without any function \(\varphi \), guarantees convergence and summable almost stability of the random Picard-Mann hybrid iteration scheme.

MSC:
47J25 Iterative procedures involving nonlinear operators
47H40 Random nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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